Tagged Questions
0
votes
1answer
45 views
Venn diagram related question
An analysis of the survey of $320$ school pupils highlighted the following facts:
• $50$ pupils live in New Town, travel to school by bus and have canteen lunch.
• $110$ pupils live in New Town ...
1
vote
1answer
41 views
question about sets
I have this as a beggining to a question:
$A\subseteq Z^2$
$$
A = \left \langle \left ( 1,7 \right );\left ( 7,2 \right );(2,3) \right \rangle = \left \{ ...
2
votes
3answers
55 views
Set Distributive Property Proof
Prove the distributive property for sets:
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
I'm not good with proofs but my understanding is that I have to prove 2 things:
(1) $A \cup (B ...
1
vote
2answers
34 views
Proper way to define this multiset operator that does a pseudo-intersection?
it's been a while since I've done anything with set theory and I'm trying to find a way to describe a certain operator.
Let's say I have two multisets:
$A = \{1,1,2,3,4\}$
$B = \{1,5,6,7\}$
How ...
0
votes
1answer
34 views
Do these two expressions mean the same?
So for a given database we have the sets Persons, Married, Women, Men and Children.
I want to express all Women who are not Children and not Married:
$$Women\setminus \left ( married \cup children ...
0
votes
3answers
47 views
Set Theory: General Intersection
How to properly prove the following:
For all integers positive integers n, if A1, A2,... and B are sets, then
0
votes
2answers
30 views
Prove that if $A\triangle B = C\triangle B$, then $A = C$
I am working with proofs in discrete math.
Help to prove:
For the sets $A$ and $B$, we define the symmetric difference of $A$ and $B$ to be $A \triangle B = (A-B)\cup(B-A).$
Prove that if $A ...
-2
votes
2answers
90 views
Proving the number of edges in the complete graph Kn
I am trying to find the number of edges in the complete graph:
$$K_n=\sum_{i=0}^{n-1} i$$
2
votes
1answer
44 views
Proving this realtion is not a transitive relation
I have trouble proving how the following statement is false:
The relation $g = \{\,(x,y)\in \Bbb R\times\Bbb R\mid y = x^2\,\}$ is transitive.
I know you have to use $yRx$, $zRy$, and $xRz$, but I'm ...
0
votes
2answers
23 views
Domain of a Relation from A to B
The latest versions of Susanna S. Epp's Discrete Mathematics book (both the First Brief Edition, and the full Applications 4th edition) define a relation from $\mathcal{A}$ to $\mathcal{B}$ as a ...
3
votes
2answers
50 views
“Tricky” wording on Congruence Modulo Question?
I'm asked for all possible values, but I can only see one. The question on my practice exam reads:
Consider the equivalence class [3] for the equivalence relation "congruence modulo 7" on $\Bbb Z$. ...
3
votes
2answers
52 views
Proving if a relation is an equivalence relation
I have been able to figure out the the distinct equivalence classes. Now I am having difficulties proving the relation IS an equivalence relation.
$F$ is the relation defined on $\Bbb Z$ as follows: ...
1
vote
3answers
74 views
Equivalence Relations: Equivalence Classes
From my basic understanding $R$ is an equivalence relation on the set $A$, which is
a relation between elements of a set that is reflexive, symmetric, and transitive.
I am not sure how to find the ...
0
votes
3answers
43 views
Relations: Reflexive, symmetric, transitive
I am having difficulties determining if this relation is reflexive, symmetric, transitive, or none of these.
Let A be the set of all strings of $0's$, $1's$, and $2's$ of length $4$. Define a ...
0
votes
2answers
35 views
Indexed unions and definitions
I've been trying to understand how to explicitly state an indexed union/intersection of sets. Here is an example:
$X$ $=$ $\bigcup S_\alpha$ where $\alpha \in I$
First of all, how would one ...
1
vote
3answers
53 views
Determine if function is well defined
I am having difficulties determining if the following function is well defined:
On certain computers the integer data type goes from $-2, 147, 483, 648$ through $2, 147, 483, 647$. Let S be the set ...
0
votes
0answers
13 views
Functions defined on General sets [duplicate]
I am learning how to determined whether a function is well defined. I am doing so by relying on two disticnt reasons that show a not well defined function: (1) There is no y that satisfies the given ...
0
votes
3answers
40 views
Function mapping
If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$?
It has been established that this is $m^n$.
How many of these are one-to-one?
I think this means that each ...
1
vote
1answer
59 views
Books/Review material on infinite cardinality for undergrad
You may have noticed me using asking many questions on Infinite Cardinalities on this fine website. Although many of the answers to my questions here were very in-depth and amazing, I just can't help ...
2
votes
1answer
32 views
Find intersection of two infinite sets
I tried searching for this problem, but I couldn't really find exactly what I was looking for. I have two sets A and B which I need to find A∩B. We are assuming that the Universal set is all Real ...
0
votes
2answers
62 views
If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable
I have a statement here I wish to prove and I would love some help on it :)
If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable
Here is my ...
1
vote
1answer
48 views
Proving every infinite set is a subset of some denumerable set and vice versa
I have 2 sets of statements that I wish to prove and I believe they are very closely related. I can prove one of them and the other I'm not so sure!
1: Every infinite set has a denumerable subset
...
3
votes
3answers
93 views
Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated!
Let $A$ be any uncountable set, and let $B$ be a countable ...
1
vote
2answers
56 views
Two-to-one functions
Let f be a function. We say that f is two-to-one provided for each $b \in\operatorname{im} f$ there are exactly two elements $a_1, a_2 \in\operatorname{dom} f$ s.t. $f(a_1) = f(a_2) = b$.
For a ...
1
vote
3answers
46 views
Cardinality of a set A is strictly less than the cardinality of the power set of A
I am trying to prove the following statement but have trouble full comprehending/going forward with some parts! Here is the statement:
If $A$ is any set, then $|A|$ $<$ $|P(A)|$
Alright, ...
1
vote
3answers
76 views
For all sets $A$ and $B$, if $A^c ⊆ B$ then $A ∪ B = U$
For all sets $A$ and $B$, if $\;A^c ⊆ B$ then $A ∪ B = U$
I am having difficulty starting to disprove an alleged set property through the use of a counterexample or if it is true then try to ...
1
vote
3answers
39 views
For all sets $A$ and B, if $B ⊆ A^c$ then $A ∩ B = ∅$
I made a Venn Diagram so I know that this is true. Now I just need some help on getting the proof right.
For all sets $A$ and B, if $B ⊆ A^c$ then $A ∩ B = ∅$
I have started the proof:
Suppose $A$ ...
0
votes
1answer
22 views
How many are not divisible by
How many numbers from 1 to 1100 are not divisible by any of 2, 5, and
11?
Self attempt:
So $|2|=550$; $|5|=220$; $|11|=100$
Using inclusion-exclusion principle, would this be the right setup?
...
1
vote
3answers
56 views
Subset question
What proportion of subsets of the digits 0-9 contain just as many odd numbers as even numbers? You need to determine how many subsets there are. Then you need to recall that you only need to consider ...
1
vote
1answer
52 views
Set of natural numbers, for which the reunion of the sums of all subsets is a set
My math is weak so I'll try to clarify my question with an example. Consider the set $\{1,2,4\}$ for which the powerset $\{ \{\}, \{1\}, \{2\}, \{4\}, \{1,2\}, \{1,4\}, \{2,4\}, \{1,2,4\} \}$.
Now, ...
5
votes
2answers
90 views
On the Definition of Posets…
In my book, the author defines posets formally in the following way:
Let $P$ be a set, and let $\le$ be a relationship on $P$ so that,
$a$. $\le$ is reflective.
$b$. $\le$ is transitive.
$c$. ...
3
votes
5answers
222 views
What does the notation $2\mathbb{Z}$ mean?
I have an assignment that is asking to define a one-to-one correspondence between the sets $2\mathbb{Z}$ and $17\mathbb{Z}$... or in other words, define some bijective function on
$$f:2\mathbb{Z}\to ...
-3
votes
1answer
63 views
element argument [closed]
1) If $5$ numbers are chosen at random from positive integers less than or equal to $2000$, what is the probability what all are divisible by 3,5, or 7?
2) Given sets $A,B$ and $C$,use an element ...
4
votes
2answers
90 views
What is the cross-product of the null set with another set? [duplicate]
Supposing that, for example, we have two sets $A$ and $B$ where $\;A = \varnothing \;$ and $\,B = \{a,b\}$.
What is the result of the cross product of those sets?
My first intuition would be to say ...
1
vote
5answers
109 views
Definition Of Symmetric Difference
The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$.
So, in set builder notation, I figured ...
1
vote
2answers
72 views
Help with sets and subsets
Show that:
if $A \subseteq C\,$ and $\,B \subseteq D,\,$ then $\,A \times B \subseteq C \times D.$
Can anyone help me with this?
1
vote
1answer
85 views
Define a $1$-$1$ onto function with domain $A$ onto the set $\{1, 2, … n\}$
Let $A = \{x^2 : x \in \mathbb{N} \text{ and } 0 \leq x^2 \leq 90\}$.
Define a 1-1 onto function with domain $A$ onto a set of the form $\{1, 2, \ldots, n\}$ to show the cardinality of $A$ is $n$.
...
1
vote
1answer
82 views
Proving Complement Laws
The problem I am working on is:
Proof the following: $A∪ \bar{A}=U$
As with all proofs, I commenced this proof by using the definition of a union:
$A∪ \bar{A} = \{x|x \in A \vee x \in ...
2
votes
1answer
87 views
Definition of the complement of a set
My book defines the complement of a set as, "Let $U$ be the universal set. The complement of the set $A$, denoted by $\bar{A}$, is the complement of $A$ with respect to $U$. Therefore, the complement ...
1
vote
2answers
82 views
Intersection and complement proof
I'm trying to prove that if $A \cap B = A \cap C$ then $A \cap \overline{B} = A \cap \overline{C}$.
I've tried several manipulations, but I can't get to it.
1
vote
1answer
126 views
Using A Venn Diagram To Illustrate Relationships
The problem I am working on is:
Use a Venn diagram to illustrate the relationship $A⊆B$ and $B⊆C$.
From my understanding, I should be drawing several Venn diagrams, corresponding to the ...
1
vote
1answer
35 views
Finding specific sets
I'm trying to calculate these particular sets given that:
$$A=\{a,c,e,h,k\}$$
$$B=\{a,b,d,e,h,i,k,l\}$$
$$C=\{a,c,e,i,m\}$$
$$A \cap B$$
$$A\cap B \cap C$$
$$A \cup B \cup C$$
$$A-B$$
...
0
votes
2answers
58 views
Math notation for an equivalence relation
What is the proper way to define a relation on $\mathbb{Z}\times\mathbb{Z}$ when $(a,b)\in\mathbb{Z}\times\mathbb{Z}$ represents $a+b$ is even?
$\mathcal{R}=\{(a,b)\in\mathbb{Z}\times\mathbb{Z} \mid ...
4
votes
2answers
141 views
Show that $(A \cap B)$ is subset of $A$
I'm having problems doing this because when I do a membership table, they do not come out to be equal.
Let $A$ and $B$ be sets.
Show that $(A \cap B)$ is subset of $A$.
2
votes
1answer
55 views
Is $A - B = \emptyset$?
$A = \{1,2,3,4,5\}, B = \{1,2,3,4,5,6,7,8\}$
$A - B =$ ?
Does that just leave me with $\emptyset$? Or do I do something with the leftover $6,7,8$?
1
vote
1answer
82 views
Show that an equivalence relation is equal to the union of its equivalence classes
Given an equivalence relation $\sim$ with equivalence classes $C_1,\dots,C_n$, show that $$\mathbin{\sim} = \bigcup_{i=1}^n(C_n\times C_n)\;.$$
I could use a hint on where to start approaching this ...
1
vote
1answer
69 views
Counting equivalence relations on set of $n$ elements
I know for a fact that the number of equivalence relations on a n element set is defined by the Bell Number.
For the case of $n=4$ the number would be then 15. But question is how do we count them?
...
0
votes
1answer
75 views
what does the inverse membership symbol means?
I know that the symbol $$ \in $$ stands for membership, but what does the symbol $$ \ni $$ stand for?
Because I know that in the set membership, one symbol stands for subset and the other ones for ...
2
votes
2answers
312 views
Reflexive , symmetric and transitive closure of a given relation
Given a relation $R = \{(x,y)\mid y=x+1\}$ and I have to find the reflexive, transitive and symmetric closure.
For reflexive, I added $y=x$ with given condition so now the relation becomes
$R = ...
3
votes
5answers
92 views
Quantification over the empty set [duplicate]
Possible Duplicate:
Why is predicate “all” as in all(SET) true if the SET is empty?
In don't quite understand this quantification over the empty set:
$\forall y \in \emptyset: Q(y)$
The ...
